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Properties

Label 4-3096e2-1.1-c1e2-0-7
Degree $4$
Conductor $9585216$
Sign $1$
Analytic cond. $611.161$
Root an. cond. $4.97209$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 4·7-s − 5·11-s + 13-s + 17-s − 2·19-s + 23-s + 2·25-s + 6·29-s + 5·31-s − 16·35-s − 8·37-s − 3·41-s + 2·43-s − 6·47-s − 2·49-s − 3·53-s + 20·55-s − 8·59-s − 8·61-s − 4·65-s − 5·67-s − 6·71-s − 6·73-s − 20·77-s − 13·83-s − 4·85-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.51·7-s − 1.50·11-s + 0.277·13-s + 0.242·17-s − 0.458·19-s + 0.208·23-s + 2/5·25-s + 1.11·29-s + 0.898·31-s − 2.70·35-s − 1.31·37-s − 0.468·41-s + 0.304·43-s − 0.875·47-s − 2/7·49-s − 0.412·53-s + 2.69·55-s − 1.04·59-s − 1.02·61-s − 0.496·65-s − 0.610·67-s − 0.712·71-s − 0.702·73-s − 2.27·77-s − 1.42·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9585216\)    =    \(2^{6} \cdot 3^{4} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(611.161\)
Root analytic conductor: \(4.97209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9585216,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
43$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_s
13$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_q
17$D_{4}$ \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_y
19$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ac
23$D_{4}$ \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_bk
29$D_{4}$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_ba
31$D_{4}$ \( 1 - 5 T + 58 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_cg
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$D_{4}$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_ai
47$D_{4}$ \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_ck
53$D_{4}$ \( 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_du
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$D_{4}$ \( 1 + 5 T + 130 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.67.f_fa
71$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_eg
73$D_{4}$ \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_ek
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$D_{4}$ \( 1 + 13 T + 198 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.83.n_hq
89$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_be
97$D_{4}$ \( 1 - 9 T + 204 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.97.aj_hw
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.248879600588993664887051487917, −8.133249369241363614326342106386, −7.88647779518510941368964205074, −7.62955953735954637908368301300, −7.08245686435232063022699142480, −6.87749131793034011106164907550, −6.06796065542319809214931627000, −5.99540707714755868799197927363, −5.13344878415857688497446370102, −5.02572421773409260157534065849, −4.57812607178267347580235484187, −4.39727024313045185016481006215, −3.66818635038367134744498014603, −3.50204828636621107696306647417, −2.69392226396855763350816612682, −2.59152454770419209770516729892, −1.46621268900372359483904829364, −1.43046628272075442915585224740, 0, 0, 1.43046628272075442915585224740, 1.46621268900372359483904829364, 2.59152454770419209770516729892, 2.69392226396855763350816612682, 3.50204828636621107696306647417, 3.66818635038367134744498014603, 4.39727024313045185016481006215, 4.57812607178267347580235484187, 5.02572421773409260157534065849, 5.13344878415857688497446370102, 5.99540707714755868799197927363, 6.06796065542319809214931627000, 6.87749131793034011106164907550, 7.08245686435232063022699142480, 7.62955953735954637908368301300, 7.88647779518510941368964205074, 8.133249369241363614326342106386, 8.248879600588993664887051487917

Graph of the $Z$-function along the critical line